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Section 9.5 Rational Exponents and Radicals

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9.5 Lecture Guide: Rational Exponents and Radicals Objective: Interpret and use rational exponents.

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Principle nth Root The principal nth root of the real number x is denoted by either or Verbally For The principal nth root is positive for all natural numbers n. For The principal nth root of 0 is 0. For If n is odd the principal nth root is negative. Examples in Radical Notation Examples in Exponential Notation

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The principal nth root of the real number x is denoted by either or Verbally Examples in Radical Notation Examples in Exponential Notation If n is even, there is no real nth root. (The nth roots will be imaginary.) For a real number. is not real number. is not a Principle nth Root

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Use the product rule for exponents to simplify the following expressions: 1. is the principal _________________________ _________________________ of x.

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Use the product rule for exponents to simplify the following expressions: 2. is the principal _________________________ _________________________ of x.

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Use the product rule for exponents to simplify the following expressions: 3. is the principal _________________________ _________________________ of x.

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Write each expression in radical notation and evaluate. 4.

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5. Write each expression in radical notation and evaluate.

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6. Write each expression in radical notation and evaluate.

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7.8. Write each expression in radical notation and evaluate.

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9. Write each expression in radical notation and evaluate.

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Rational Exponents For a real number x and rational numbers m and n : Algebraically Examples in Radical Notation Examples in Exponential Notation If is a real number* or

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Rational Exponents For a real number x and rational numbers m and n : Algebraically Examples in Radical Notation Examples in Exponential Notation If is a real number* *Ifand n is even, then is not a real number.

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Write each expression in radical notation and evaluate. 10.

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11. Write each expression in radical notation and evaluate.

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12. Write each expression in radical notation and evaluate.

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13. Write each expression in radical notation and evaluate.

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14. Write each expression in radical notation and evaluate.

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15. Write each expression in radical notation and evaluate.

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Use a graphing calculator to approximate each expression to the nearest hundredth. 16.

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Use a graphing calculator to approximate each expression to the nearest hundredth. 17.

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Use a graphing calculator to approximate each expression to the nearest hundredth. 18.

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Use a graphing calculator to approximate each expression to the nearest hundredth. 19.

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Exponent Rules You should be familiar with the properties of integer exponents. Note that these properties apply to all real exponents, including rational exponents. Let m, n, x, y, x m, x n, y m, and y n be real numbers Product rule: Product to a Power: Quotient rule: Power rule: Quotient to a Power: Negative power: with and

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Simplify each expression. Assume that x is a positive real number. 20.

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Simplify each expression. Assume that x is a positive real number. 21.

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Simplify each expression. Assume that x is a positive real number. 22.

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Simplify each expression. Assume that x is a positive real number. 23.

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Simplify each expression. Assume that x and y are positive real numbers. 24.

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Simplify each expression. Assume that x and y are positive real numbers. 25.

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Simplify each expression. Assume that x and y are positive real numbers. 26.

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Simplify each expression. Assume that x and y are positive real numbers. 27.

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Simplify each expression. Assume that x and y are positive real numbers. 28.

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Simplify each expression. Assume that x and y are positive real numbers. 29.

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Simplify each expression. Assume that x and y are positive real numbers. 30.

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Simplify each expression. Assume that x and y are positive real numbers. 31.

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Simplify each expression. 32.

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Simplify each expression. 33.

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Simplify each expression. 34.

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Simplify each expression. 35.

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